A047890 Number of permutations in S_n with longest increasing subsequence of length <= 5.

1, 1, 2, 6, 24, 120, 719, 5003, 39429, 344837, 3291590, 33835114, 370531683, 4285711539, 51990339068, 657723056000, 8636422912277, 117241501095189, 1639974912709122, 23570308719710838, 347217077020664880, 5231433025400049936, 80466744544235325387

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..735

F. Bergeron and F. Gascon, Counting Young tableaux of bounded height, J. Integer Sequences, Vol. 3 (2000), #00.1.7.

Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, and Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.

Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, and Jean-Marie Maillard, Stieltjes Moment Sequences for Pattern-Avoiding Permutations, Electron. J. Combin., Vol. 27, No. 4 (2020), p. 30.

Shalosh B. Ekhad, Nathaniel Shar, and Doron Zeilberger, The number of 1...d-avoiding permutations of length d+r for SYMBOLIC d but numeric r, arXiv:1504.02513, 2015.

Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory Ser. A 53 (1990), no. 2, 257-285.

Permutation Pattern Avoidance Library (PermPAL), 012345

Nathaniel Shar, Experimental methods in permutation patterns and bijective proof, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.

Index entries for sequences related to Young tableaux.

Formula

a(n) ~ 9 * 5^(2*n + 25/2) / (512 * n^12 * Pi^2). - Vaclav Kotesovec, Sep 10 2014

Maple

h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
      +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= proc(n, i, l)
      `if`(n=0 or i=1, h([l[], 1$n])^2, `if`(i<1, 0,
       add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
    end:
a:= n-> g(n, 5, []):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 10 2012
# Alternative:
a:= proc(n) option remember; `if`(n<6, n!, ((-375+400*n+843*n^2
       +322*n^3+35*n^4)*a(n-1) +225*(n-1)^2*(n-2)^2*a(n-3)
       -(259*n^2+622*n+45)*(n-1)^2*a(n-2))/ ((n+6)^2*(n+4)^2))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 26 2012

Mathematica

h[l_] := With[{n = Length[l]}, Sum[i, {i, l}]!/Product[Product[1+l[[i]]-j+Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]]; g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Array[1&, n]]]^2, If[i<1, 0, Sum[g[n-i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]]; a[n_, k_] := If[k >= n, n!, g[n, k, {}]]; Table[a[n, 5], {n, 1, 30}] (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)

Crossrefs

A column of A047888. Cf. A005802, A052399.

Column k=5 of A214015.

Sequence in context: A364231 A364234 A052398 * A297204 A071088 A177533

Adjacent sequences: A047887 A047888 A047889 * A047891 A047892 A047893

Keyword

nonn,easy

Author

Eric Rains (rains(AT)caltech.edu), N. J. A. Sloane

Extensions

More terms from Naohiro Nomoto, Mar 01 2002

More terms from Alois P. Heinz, Apr 10 2012

Status

approved